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arXiv:quant-ph/0404074 · Journal of Physics A Mathematical and General

The Wigner function associated with the Rogers-Szegö polynomials

D. Galetti, Salomon Sylvain Mizrahi, M. Ruzzi

Published Dec 2, 2004• 4 citations
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Abstract

A Wigner function associated with the Rogers-Szegö polynomials is proposed and its properties are discussed. It is shown that from such a Wigner function it is possible to obtain well-behaved probability distribution functions for both angle and action variables, defined on the compact support −π ⩽ θ < π, and for m ⩾ 0, respectively. The width of the angle probability density is governed by the free parameter q characterizing the polynomials.

DOI
10.1088/0305-4470/37/50/l01

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A Wigner function associated with the Rogers-Szegö polynomials is proposed and its properties are discussed. It is shown that from such a Wigner function it is possible to obtain well-behaved probability distribution functions for both angle and action variables, defined on the compact support −π ⩽ θ < π, and for m ⩾ 0, respectively. The width of the angle probability density is governed by the free parameter q characterizing the polynomials.
By means of a well-established algebraic framework, Rogers–Szegö functions associated with a circular geometry in the complex plane are introduced in the context of q -special functions, and their properties are discussed in detail. The eigenfunctions related to the coherent and phase states emerge from this formalism as infinite expansions of Rogers–Szegö functions, the coefficients being determined through proper eigenvalue equations in each situation. Furthermore, a complementary study on the Robertson–Schrödinger and symmetrical uncertainty relations for the cosine, sine and nondeformed number operators is also conducted, corroborating, in this way, certain features of q -deformed coherent states.
By means of a well-established algebraic framework, Rogers-Szego functions associated with a circular geometry in the complex plane are introduced in the context of q-special functions, and their properties are discussed in details. The eigenfunctions related to the coherent and phase states emerge from this formalism as infinite expansions of Rogers-Szego functions, the coefficients being determined through proper eigenvalue equations in each situation. Furthermore, a complementary study on the Robertson-Schrodinger and symmetrical uncertainty relations for the cosine, sine and nondeformed number operators is also conducted, corroborating, in this way, certain features of q-deformed coherent states.
We discuss some results from q-series that can account for the foundations for the introduction of orthogonal polynomials on the circle and on the line, namely the Rogers-Szegö and Stieltjes-Wigert polynomials. These polynomials are explicitly written and their orthogonality is verified. Explicit realizations of the raising and lowering operators for these polynomials are introduced in analogy to those of the Hermite polynomials that are shown to obey the q-commutation relations associated with the q-deformed harmonic oscillator.
We study a generalization of the Wigner function to arbitrary tuples of hermitian operators. We show that for any collection of hermitian operators A1...An , and any quantum state there is a unique joint distribution on R^n, with the property that the marginals of all linear combinations of the operators coincide with their quantum counterpart. In other words, we consider the inverse Radon transform of the exact quantum probability distributions of all linear combinations. We call it the Wigner distribution, because for position and momentum this property defines the standard Wigner function. We discuss the application to finite dimensional systems, establish many basic properties and illustrate these by examples. The properties include the support, the location of singularities, positivity, the behavior under symmetry groups, and informational completeness.
We study a generalization of the Wigner function to arbitrary tuples of Hermitian operators. We show that for any collection of Hermitian operators A1, …, An and any quantum state, there is a unique joint distribution on Rn with the property that the marginals of all linear combinations of the Ak coincide with their quantum counterparts. In other words, we consider the inverse Radon transform of the exact quantum probability distributions of all linear combinations. We call it the Wigner distribution because for position and momentum, this property defines the standard Wigner function. We discuss the application to finite dimensional systems, establish many basic properties, and illustrate these by examples. The properties include the support, the location of singularities, positivity, the behavior under symmetry groups, and informational completeness.
We derive explicit expressions for the Wigner function of wave functions in DDD dimensions which depend on the hyperradius---that is, of sss waves. They are based either on the position or the momentum representation of the sss wave. The corresponding Wigner function depends on three variables: the absolute value of the DDD-dimensional position and momentum vectors and the angle between them. We illustrate these expressions by calculating and discussing the Wigner functions of an elementary sss wave and the energy eigenfunction of a free particle.